Question

For what value of a, the divisions of two polynomials (ax³ + 3x² - 3) and (2x³ - 5x + a) by (x - 4) give the same remainder.
Let us calculate and write it. ​

Answer 1

$\huge\bf\underline{Answer}$

The value of a is 1 respectively.

Solution:-

In the first case,

f(x)= ax³ + 3x² - 3 when divided by the linear polynomial (x - 4)

At first, we will find out the zero of the polynomial (x - 4):-

=> x - 4 = 0

=> x = 4

From the remainder theorem,

The required remainder = f(4)

Putting the value of x,

= ax³ + 3x² - 3

= a.(4)³ + 3.(4)² - 3

= 64a + 48 - 3

= 64a + 45 ........(i)

In the second case,

g(x) = 2x³ - 5x + a when divided by the linear polynomial (x - 4)

Zero of the polynomial (x - 4) = 4

Putting the value of x

= 2x³ - 5x + a

= 2.(4)³ - 5.4 + a

= 2.64 - 20 + a

= 128 - 20 + a

= a + 108 ........(ii)

By comparing eqn.(i) & (ii), we get:-

=> 64a + 45 = a + 108

=> 64a - a = 108 - 45

=> 63a = 63

=> a = 63/63

=> a = 1

Hence, the value of a is 1.

Verification:-

Putting the value of a and x in the polynomial f(x), we get:-

= ax³ + 3x² - 3

= 1.(4)³ + 3.(4)² - 3

= 64 + 48 - 3

= 109

Putting the value of a and x in the polynomial g(x), we get:-

= 2x³ - 5x + a

= 2.(4)³ - 5.4 + 1

= 128 - 20 + 1

= 109

Therefore, the remainders arethe same.

Hence, proved.

Answer 2

Answer:

Value of a = 1

Step-by-step explanation:

Given Problem:

For what value of a, the divisions of two polynomials (ax³ + 3x² - 3) and (2x³ - 5x + a) by (x - 4) give the same remainder.  

Let us calculate and write it. ​

Solution:

To Find:

Value of a.

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Method:

Given that polynomials are:

$\implies\ ax^3 + 3x^2 - 3\; and \ 2x^3- 5x +a$

It is also given that these two polynomials leave the same remainder when divided by $( x - 4)$

So,

$\implies\ a(4)^3 + 3 (4)^2 - 3 = 2 (4)^3 - 5 (4)+a$

$\implies\ 64a + 48 -3 = 108 - 45$

$\implies\ 63a = 63$

$\implies\ a = 1$